Simple model for the bottleneck effect in isotropic turbulence based on Kolmogorov's hypotheses

We propose a simple model for the bottleneck effect in homogeneous isotropic turbulence as a bump in the compensated energy spectrum, based on Kolmogorov's hypotheses (of 1941 and 1962). The model of the longitudinal structure function consists of two quadratic functions representing large- and small-scale motions. The model parameters are derived from the asymptotic behavior of the structure function. The Kolmogorov and intermittency constants are fitted from direct numerical simulation (DNS) and experimental data. From the model, the height of the spectral bump in the compensated spectrum has a power-law $R_{\lambda }^{-0.0426}$, and the bump location scaled by the Kolmogorov scale is 0.153, which generally agree with various DNS results at moderate and large Reynolds numbers $R_{\lambda }$. Moreover, we derive that the incorporation of the intermittency exponent into the model leads to the decaying power law of the bump height with $R_{\lambda }$.

Figure 1

Normalized (a) longitudinal structure function and (b) energy spectrum at a wide range of $R_{\lambda }$. Lines: model in Eq. (11) with $C_2=2.2$ and $\mu =0.25$; symbols: DNSs [34, 35].

Figure 2

Comparison of compensated three-dimensional energy spectra with $R_{\lambda }=140$, 240, 400, 650, and 1000. Symbols: DNS [5]; lines: model in Eq. (11) with $C_2=2.2$ and $\mu =0.25$.

Figure 3

Comparisons of (a) peak heights and (b) locations of spectral bumps from the model ($▿$) and DNS data ($\circ$: Ref. [3]; $\times$: Ref. [5]; $◊$: Ref. [34]; $▵$: Ref. [6]) with their linear fits or averages (lines). The model results are calculated from Eq. (11) with $C_2=2.2$ and $\mu =0.25$.

Figure 4

Effects of the variation in $C_2$ on modeling the (a) height and (b) location of the spectral bumps of $\mathrm{\Psi }\left(k\eta \right)$ at a range of $R_{\lambda }$ with $\mu =0.25$. Lines in (a): linear fits of modeling results with $C_2=2.0,2.05,...,2.40$, from bottom to top; lines in (b): averages of modeling results with $C_2=1.6,1.7,...,2.4$, from top to bottom; symbols: DNS results ($\circ$: Ref. [3], $\times$: Ref. [5], $◊$: Ref. [34], and $▵$: Ref. [6]).

Figure 5

Effects of the variation in $\mu$ on modeling the (a) height and (b) location of the spectral bumps of $\mathrm{\Psi }\left(k\eta \right)$ at a range of $R_{\lambda }$ with $C_2=2.2$. Lines in (a): linear fits of modeling results with $\mu =0.20,0.22,...,0.30$, from top to bottom; lines in (b): averages of modeling results with $\mu =0.20,0.22,...,0.30$, from top to bottom; symbols: DNS results ($\circ$: Ref. [3], $\times$: Ref. [5], $◊$: Ref. [34], and $▵$: Ref. [6]).

Figure 6

Comparison of compensated three-dimensional energy spectra with $R_{\lambda }=140$, 240, 400, 650, and 1000. Symbols: DNS [5]; lines: model in Eq. (B1) with $p=1.8,q=2,C_2=2.3$, and $\mu =0.25$.

Figure 7

Comparisons of (a) peak heights and (b) locations of spectral bumps from the model ($▿$) and DNS data ($\circ$: Ref. [3]; $\times$: Ref. [5]; $◊$: Ref. [34]; $▵$: Ref. [6]) with their linear fits or averages (lines). The model results are calculated from Eq. (B1) with $p=1.8$, $q=2$, $C_2=2.3$, $\mu =0.25$, and Eq. (8).